Independent families in Boolean algebras with some separation properties

Piotr Koszmider*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size c, the size of the continuum. This improves a result of Argyros from the 1980s, which asserted the existence of an uncountable independent family. In fact, we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone space KA of all such Boolean algebras A contains a copy of the Čech-Stone compactification of the integers βℕ and the Banach space C(KA has l as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.

Original languageEnglish
Pages (from-to)305-312
Number of pages8
JournalAlgebra Universalis
Volume69
Issue number4
DOIs
StatePublished - Jun 2013

Keywords

  • 46B10
  • 54D
  • Efimov's problem
  • Grothendieck property
  • independent families
  • Primary: 06E
  • Secondary: 46E15
  • subsequential completeness property

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